Environmental Engineering Reference
In-Depth Information
Within the scope of a linear or linearised calculation of the hydrodynamic forces for a
floating body, it is possible to split the perturbation potential into diffraction and
radiation problems:
Þ
X
6
j
¼
1
_
F
S
x
ð
;
y
;
z
;
t
Þ F
diff
x
ð
;
y
;
z
;
t
s
j0
F
j
x
ð
;
y
;
z
;
t
Þ
where
F
diff
perturbation potential for the flow around the body held in the primary wave
F
j
potential of the flow field that results from the enforced motion of the body in
direction “j” with velocity amplitude “1” in the originally smooth water
s
j0
complex amplitude of velocity of body motion in direction j (j
¼
1, 2, 3:
translations; j
¼
4, 5, 6: rotations)
The superposition of the radiation and these solutions to the boundary value
problems of the diffraction leads to a solution for the floating body according to
potential theory.
Only the diffraction problem has to be solved for offshore structures permanently
anchored to the seabed. The total potential describing the interaction between wave and
structure is therefore expressed as follows:
ð Þ
Both potentials are defined between still water level (z
¼
0) and seabed (z
¼
d). The
perturbation potential oscillates harmonically with the frequency of the incident wave
(
F
ð
x
;
y
;
z
;
t
Þ F
0
x
ð
;
y
;
z
;
t
Þ F
diff
x
;
y
;
z
;
v¼
2
p
f). Therefore, it follows that
Þ
e
i
v
t
F
diff
x
ð
;
;
;
Þ w
diff
x
ð
;
;
y
z
t
y
z
where
w
diff
(x, y, z) steady-state part of perturbation potential
The perturbation potential must always satisfy the Laplace differential equation:
ð Þ
0
Furthermore, it must satisfy the following boundary conditions:
- the combined linearised boundary condition at the surface of the water
DF
diff
x
ð
;
y
;
z
;
t
Þ
0 r
Dw
diff
x
;
y
;
z
;
2
@
F
diff
@
þ
g
@F
diff
@
w
diff
þ
g
@F
diff
@
2
z
¼
0 r
v
z
¼
0
t
2
- and the kinematic boundary condition at the seabed
@F
diff
@
z
¼
0 r
@w
diff
z
¼
0
@
